Just cover yourself. If you go privately to your teacher, he will understand because he knows it takes a LOT for you to admit to an error. Nowadays, school pressures are insane. You should be able to clear your conscience once you speak privately to your teacher without having to implicate others.
It shouldn’t show up on your transcript, or your counselor’s record, if you just speak privately to the teacher. If you delay it, the teacher may not be as sympathetic later. You’ve learned from it, and it’s time to move on from this lesson.